Undrained stability of dual tunnels in layered soils with different strength

The stability of dual circle tunnels buried in layered soils with different shear strengths was investigated by using finite element limit analysis (FELA). The emphasis of this study is in quantitating that the existing tunnels affects the newly-built one, and design suggestions have been provided especially in the optimum location of construction. By imposing a FELA modelling, the variation trends of undrained bearing capacity with different influential factors, including the horizontal distance, vertical distance, the thickness of the top layer, the shear strength ratio of the layered soil, were further investigated. It is concluded that there exists an inclination-fixed worst-band, in which there would be a worst undrained stability once the bottom tunnel was constructed in the band. It is interested that the inclination seems constant by varying several factors but the horizontal distance would be changed with the soil properties. In addition, three patterns of collapse were summarized.

www.nature.com/scientificreports/ bottom layer is S/D which is fixed as 0.5 in this study to simplify calculations. For homogeneous soil, the bearing capacity can be expressed as a stability factor N c = σ s /c u 13 . Hence, the stability factor N c can be defined as follows where σ s is the surcharge load, c u1 , c u2 and γ are the undrained shear strength of top layer, bottom layer and unit weight of soil, which follows the Tresca failure criterion 3,[11][12][13]18,19 . The unit weight is fixed as 20 kN/m 3 . And cases of c u1 /c u2 = 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5 are investigated in this study.

FELA modelling
As an accurate and efficient method, FELA was used to address geotechnical stability issues [1][2][3]7,[9][10][11][12][13][14][15][16][17][18][19][20][21][22] . FELA method can acquire accurate ultimate loads by a combination of the limit theorems of plasticity and finite elements 23 . Strictly close predictions of lower bound (LB) and upper bound (UB) results can be obtained by reasonable construction of the statically admissible stress field and kinematically admissible velocity field, respectively. In view of the superiority of FELA method, a computational software Optum G2 24 is employed in this study to investigate the ultimate bearing capacity and failure mechanisms of dual tunnels in layered soil with different strength. This program can refine precise mesh on the strength of the adaptive meshing technique. In this study, the number of initial mesh and finial mesh is set as 1000 and 5000 to obtain the accurate results and reasonable operation time, and the iteration of mesh refinement is set as 3 steps. The lateral and bottom boundaries are fixed in all directions to follow previous literature 1,2 . The size of soil field was set as 40B*20B which is large enough to avoid the boundary effect. And the general view of this FELA model is depicted in Fig. 2.
To ensure the reliability of the FELA modelling, the average value of UB and LB FELA results are compared with previous literature 2 and FEM method. Figure 3a shows the cases of dual voids in single layer soil at different depth. It can be seen that the results of FELA agree well with Xiao's results. Figure 3b shows the comparison   It is well known that one of the superiorities for FELA method is the adaptive meshing technique, providing a more reasonable mesh to obtain more accurate results than the FEM method. There would be discrepancies caused by different meshing methods, and a relatively small discrepancy (less than 5.7%) proves the reliability of FELA modelling.

Results and discussions
The effect of influential factors presented in Eq. (1) on N c are investigated as follows.
Effect of the shear strength ratio c u1 /c u2 . Figures 4 and 5 show the variation trends of undrained bearing capacity with different c u1 /c u2 and horizontal/vertical distance between the double tunnels. It is obvious that for all cases, the bearing capacity keeps constant before c u1 /c u2 = 1, indicating that the deeper tunnel has no effect on the bearing capacity. It reasons that the bottom layer is stiff enough to resist the distributed load which can lead the collapse of the shallower tunnel. When c u1 /c u2 > 1, the bearing capacity decreases with the increase of c u1 /c u2 , indicating that the influence of soft bottom layer becomes significant. To sketch the trend accurately,  Fig. 4c), the knee point is at c u1 /c u2 = 1.5, which is smaller than the c u1 /c u2 of knee point for cases of H/D = 2 (depicted in Fig. 4b). Furthermore, it can be seen that the bearing capacity in Fig. 4 decrease with the increase of c u1 /c u2 , and all curves tend to overlap with the increase of c u1 /c u2 . It indicates that the effect of X/D becomes  www.nature.com/scientificreports/ a less value of c u1 /c u2 . And the scatter of group of curves in Fig. 5 raises with the increase of c u1 /c u2 . It reasons that the deeper the tunnel is, the greater the soil load it bears. This trend becomes more significant with the increase of c u1 /c u2 . And the soil weight would lead to a negative value of bearing capacity. The negative value means that the tunnel requires extra lining to maintain its stability.
Effect of the vertical distance of tunnels Y/D. Figure 6 presents the variation trend of bearing capacity with different Y/D. It can be seen that for nearly all cases, the depth of bottom tunnel affects the bearing capacity linearly. As expected, all curves of Fig. 6 show a downtrend with the increase of Y/D, which is caused by the large unit weight of soil. And the curves of Fig. 6b are highly coincidence, it illustrates that the influence of X/D becomes weaker with greater c u1 /c u2 and less H/D. Furthermore, it is interesting that there are some cross points in Fig. 6. It indicates that the bearing capacity would not increase with the increase of X/D monotonously. The detailed investigation of this phenomenon would be presented in the next section.  Table 1 for a better understanding, obviously, the intervals between adjacent curve are all equal to 0.5, further revealing the existence of the worst-band. Based on these observations, helpful suggestions can be drawn that if there would be a newly-built tunnel near an existing tunnel, the site selection of the newly-built tunnel should locate away from the worst-band to guarantee an optimum bearing capacity.

Effect
In addition, it can be observed from Figs. 7, 8 and 9 that the curves show a downtrend with the increase of X/D, and the bearing capacity of X/D = 0 is the greatest. Generally, the closer the distance between dual tunnels/ voids is, the greater the interaction between them becomes. And the interaction between dual tunnels can weaken the stability of the system, which is inconsistent with the common sense. In view of this inconsistency, analyses of failure mechanisms would be presented in next section to explain this phenomenon.

Failure mechanism
For deeper insight into the bearing capacity of whole system, several typical failure patterns are discussed in this section. Figure 11 shows the failure mechanism with c u1 /c u2 = 0.5, X/D = 2, Y/D = 4. It is obvious that the failure zone is confined into the top layer, and the failure pattern is the typical side wall failure of single tunnel. The reason for this failure pattern is when c u1 /c u2 < 1, the bottom layer is stiff enough to avoid the collapse before the top tunnel is totally collapsed. That is to say, the failure pattern and bearing capacity would keep invariable if the c u1 /c u2 < 1, which corresponds to tendencies of Figs. 4 and 5. Figure 12 shows a special failure pattern with c u1 /c u2 = 3, X/D = 0, Y/D = 4. It can be seen that there exists similar triangular wedge above each tunnel. But the reasons for these two triangular wedge are different. For the top wedge, it is a typical failure pattern of a single void under surcharge loading, which means that the top of the tunnel is under pressure stress. And the top compression collapse is the primary cause of the top tunnel failure. As to the occurrence of bottom wedge, it reasons that the tunnel in the top stiffer soil bear the surcharge loading and prevent a deeper load transmission to the bottom layer. Hence, it can be observed that there has no failure curve between these two tunnels, indicating that the collapse of top tunnel is depended upon the surcharge loading but not relies on the bottom tunnel. In addition, it can be seen that the deformation around the bottom tunnel is distinct. Namely, the circumferential squeezing to the bottom tunnel is the dominating collapse of this system. Figure 13 presents the failure mechanism with c u1 /c u2 = 3, X/D = 1, Y/D = 4. It is obvious that the primary collapse is also the circumferential squeezing to the bottom tunnel. However, with the shift of the bottom tunnel, it  Figure 14 shows the worst case of Y/D = 4. It can be found that there are two features which cause the worst stability of whole system: (1) an obvious failure curve between the dual tunnels, which means the bottom tunnel (2) an obvious failure curve between the top tunnel and the ground surface, it means that the surcharge loading also has great influence on the top tunnel. And the combination of strong interaction between the tunnels and the great influence of surcharge loading lead to a large deformation of the top tunnel, which would cause the worst stability. Figure 15 shows the failure mechanisms with c u1 /c u2 = 3, Y/D = 4, X/D = 7 and 10. It can be found from Figs. 14 and 15a that the interaction between the tunnels becomes weaker with the increase of X/D. The failure curve www.nature.com/scientificreports/ between the top tunnel and the ground surface also lighter than the worst case (depicted in Fig. 14). And the circumferential squeezing to the bottom tunnel becomes to the primary collapse again. With the farther shift of bottom tunnel (X/D = 10), it can be seen from Fig. 15b that there is no failure curve between the ground and the top tunnel, which illustrates that the surcharge loading has no influence on the top tunnel. And all the deformation of top tunnel is caused by the collapse of bottom tunnel, and the interaction between the dual tunnels is www.nature.com/scientificreports/ very weak. In view of this, it can be predicted that the circumferential squeezing to the bottom tunnel would be the only collapse type if the horizontal distance between the tunnels is far enough. It is worth noting that there are three typical failure patterns can be summarized to describe the interaction between the dual tunnels. Figure 12 depicts a situation that there has no interaction between the tunnels for cases of a much close horizontal distance between the dual tunnels, which is defined as 'independent double tunnel    Figure 15 shows a failure pattern that the interaction between tunnels almost disappear with the increase of horizontal distance between tunnels, which defined as 'primary bottom tunnel failure' . And the variation of failure mechanisms presented in Figs

Conclusions
This study employs FELA to investigate the undrained stability of dual tunnels locate in layered soils. Several influential factors have been investigated, including the thickness of top layer (H/D), the horizontal and vertical distance between the dual tunnels (X/D and Y/D) and the shear strength ratio (c u1 /c u2 ). Representative failure patterns are also discussed for deeper insight into the bearing capacity of whole system. Based on these results from FELA, some conclusions can be drawn as follows (it should be noted that the following conclusions are all based on c u1 /c u2 ≥ 1, because the bearing capacity and the failure mechanism are invariable for cases of c u1 /c u2 < 1): (1) The curve of stability shows a downtrend initially than uptrend with the increase of horizontal distance between the dual tunnels; (2) There is an angle-fixed worst-band; the closer the distance between the bottom tunnel and the worst-band is, the worse the stability of the whole system becomes; (3) The worst-band would move farther away from the top tunnel with the increase of H/D and c u1 /c u2 ; (4) Three typical failure patterns are summarized to reveal the interaction between the two tunnels. The patterns can further verify the variation of bearing capacity.

Data availability
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